You could spend your entire life watching anime (24/7/365) and you would still be missing anime.
Serial Experiments Lain
Now and Then, Here and There
Tengen Toppa Gurren Lagaan
Welcome to the NHK!
I'd recommend checking out Gintama first. The first season is alright but it becomes a lot better later on. And everyone's seen Elfen Lied, it's pretty much a mainstream anime.